Stability verification and timing contract synthesis for linear impulsive systems using reachability analysis

This paper deals with stability analysis for a class of linear impulsive systems subject to a timing contract specifying bounds on the time between two consecutive impulses. We consider the problem of stability verification, which consists in proving stability for a particular timing contract, and the problem of timing contract synthesis, which consists in synthesizing a set of timing contracts that guarantee the stability of the linear impulsive system. Our approach is based on a reformulation using parameterized difference inclusions. We derive theoretical necessary and sufficient conditions for stability based on the propagation of a set by the system dynamics. For linear impulsive systems, when using approximate reachability analysis, this allows us to state a sufficient condition for stability and to design a stability verification algorithm. We then propose an approach to timing contract synthesis, which exploits the monotonicity of stability with respect to timing contract parameters to design an algorithm based on adaptive sampling of the parameter space. Several examples are provided, which allow us to compare our algorithm with several existing techniques, and show the effectiveness of our approach.